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Optimal Exploration of Small Rings

Optimal Exploration of Small Rings. Stéphane Devismes VERIMAG UMR 5104 Univ . Joseph Fourier Grenoble, France. Talk by Franck Petit , Univ . Pierre et Marie Curie - Paris 6, France. Context. Autonomous. : No central authority. Anonymous. : Undistinguishable . Oblivious.

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Optimal Exploration of Small Rings

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  1. Optimal Exploration of Small Rings Stéphane Devismes VERIMAG UMR 5104Univ. Joseph FourierGrenoble, France Talk by Franck Petit, Univ. Pierre et Marie Curie - Paris 6, France

  2. Context • Autonomous • : No central authority • Anonymous • : Undistinguishable • Oblivious • : No mean to know the past • Disoriented • : No mean to agree on a common direction or orientation A team of k “weak” robots evolving into a ring of n nodes

  3. Context • Atomicity • : In every configuration, each robot is located at exactly one node • Weak Multiplicity • : In every configuration, each node may contain some robots • (a robot cannot detect the exact number of robots located at each node but it is able to detect if there are zero, one, or more) A team of k “weak” robots evolving into a ring of n nodes

  4. Context • SSM • : In every configuration, k’ robots are activated (0 < k’ ≤ k) • The k’ activated robots execute the cycle: • Look • : Instantaneous snapshot with multiplicity detection • : Based on this observation, decides to either stay idle or move to one of the neighboring nodes • Compute • Move • : Move toward its destination A team of k “weak” robots evolving into a ring of n nodes

  5. Problem: Exploration • Starting from a configuration where no two robotsare located at the same node: • Performance:Number of robots (k<n) Exploration:Each node must be visited by at least one robot Termination:Eventually, every robot stays idle

  6. Relatedworks (Deterministic) Tree networksΩ(n) robots are necessary in generalA deterministicalgorithmwithO(log n/log log n) robots, assumingthatΔ ≤ 3[Flocchini, Ilcinkas, Pelc, Santoro, SIROCCO 08] Ring networksΘ(log n) robots are necessary and sufficient, providedthatn and k are coprimeA deterministicalgorithm for k ≥ 17[Flocchini, Ilcinkas, Pelc, Santoro, OPODIS 07]

  7. Relatedworks (Probabilistic) • Ring networks [Devismes, Petit, Tixeuil, SIROCCO 2010] • 4 robots are necessary • For ring of sizen>8, 4 robots are sufficient to solve the problem

  8. Contribution Question.Are4 probabilistic robots necessary and sufficient to explore any ring of any size n ? • Remark. • The problemis not defined for n < 4 • For n = 4, no algorithmrequired • Contribution. • Algorithm for 5 ≤ n ≤ 8 • Corollary: 4 probabilistic robots are necessaryand • sufficient to explore any ring of any size n

  9. Definitions Tower.A node with at least two robots. k ≥ 2

  10. Definitions Segment.A maximal non-empty elementary path of occupied nodes. a 2-segment A 1-segment

  11. Definitions Hole.A maximal non-empty elementary path of free nodes. 1 hole of length 3 A 1-hole

  12. Definitions Arrow. A 1-segment, followed by a hole, a tower, and a 1-segment. Head Tail 1 arrow of length 2

  13. Definitions Arrow. A 1-segment, followed by a hole, a tower, and a 1-segment. Primaryarrow

  14. Definitions Arrow. A 1-segment, followed by a hole, a tower, and a 1-segment. final arrow

  15. Algorithm: Overview Let start with phase II and III, it’s easier … • 3 main steps: • Phase I: Initial configuration 4-segment • Invariant: no arrow • Phase II: 4-segmentprimaryarrow • Invariant: 4-segment or primaryarrow • Phase III: Primaryarrowfinalarrow • Invariant: increasingarrow • (2 special cases)

  16. Algorithm: Phase II Probabilistic moves • Phase II: 4-segmentprimaryarrow • Invariant: 4-segment or primaryarrow

  17. Algorithm: Phase II Primaryarrow • Phase II: 4-segmentprimaryarrow • Invariant: 4-segment or primaryarrow

  18. Algorithm: Phase III Deterministic move • Phase III: Primaryarrowfinalarrow • Invariant: increasingarrow

  19. Algorithm: Phase III • Phase III: Primaryarrowfinalarrow • Invariant: increasingarrow

  20. Algorithm: Phase III • Phase III: Primaryarrowfinalarrow • Invariant: increasingarrow

  21. Algorithm: Phase III • Phase III: Primaryarrowfinalarrow • Invariant: increasingarrow

  22. Algorithm: Phase III Termination • Phase III: Primaryarrowfinalarrow • Invariant: increasingarrow

  23. Algorithm: Back to Phase I • Phase I: Initial configuration 4-segment • Invariant: no arrow • Principle: • No symmetry: Deterministic moves • Symmetry: Probabilistic or deterministic moves

  24. Phase I: no symmetry • There exists a unique largest segment S: • move towardS following the shortestneighboringhole

  25. Phase I: no symmetry Ambiguity: Decisiontaken by an adversary • There exists a unique largest segment S: • move towardS following the shortestneighboringhole

  26. Phase I: no symmetry Ambiguity: Decisiontaken by an adversary • There exists a unique largest segment S: • move towardS following the shortestneighboringhole

  27. Phase I: no symmetry • There exists a unique largest segment S: • move towardS following the shortestneighboringhole

  28. Phase I: symmetry Case by Case Study

  29. Phase I: n = 5 • No symmetry • Initial configuration: a 4-segment • Phase I & II

  30. Phase I: n = 6 The 2 special cases Stop Stop Only one symmetryisinitially possible

  31. Phase I: n = 7 Only one symmetryisinitially possible

  32. Phase I: n = 8 (c) (b) (a) Threesymmetries are initially possible:

  33. Phase I: n = 8, Case (a) Case (c)

  34. Phase I: n = 8, Case (b) Case (c)

  35. Phase I: n = 8, Case (c) (c) Reallycomplex!!! See the paper…

  36. Conclusion • General Result: • 4probabilistic robots are necessary and sufficient to solve the exploration of anyanonymous ring • Future works: • Convergence time (experimentalresult:O(n) moves) • Full asynchronous model • Other (regular) topologies

  37. Conclusion Thankyou.

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